ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

4. Algebraic Geometry

1991 MS Classification: 14E09, 14E25 Shpilrain, Vladimir and Yu, Jie-Tai, University of Hong Kong Embeddings of curves in the plane Let K[x, y] be the polynomial algebra in two variables over a field K of characteristic 0. Here we contribute toward a classification of two-variable polynomials by classifying (up to an automorphism of K[x, y]) polynomials of the form x^{n} + y^{m} + ∑_{im+jn≤mn} c_{ij}x^{i}y^{j}, c_{ij} ∈ K. Our classification has several applications to the study of embeddings of algebraic curves in the plane. In particular, we show that for any k ≥ 2, there is an irreducible curve which has at least k inequivalent embeddings in C^{2}. Also, upon combining our method with a well-known theorem of Zaidenberg and Lin, we show that one can decide “almost“ just by inspection whether or not a polynomial fiber {p(x, y) = 0} is an irreducible simply connected curve.